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Representations of the (-2,3,7)-Pretzel Knot and Orderability of Dehn Surgeries

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Representations of the (-2,3,7)-Pretzel Knot and Orderability of Dehn Surgeries Representations of the (-2,3,7)-Pretzel Knot and Orderability of Dehn Surgeries Konstantinos Varvarezos November 27, 2019 Abstract We construct a 1-parameter family of SL2(R) representations of the pretzel knot P (−2; 3; 7). As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6. Furthermore, we discuss a family of knots and exhibit similar orderability results for a few other examples. 1 Introduction This paper studies the character variety of a certain knot group in view of the relationship between PSL^2(R) representations and the orderability of Dehn surgeries on the knot. This is of interest because of an outstanding conjectured relationship between orderability and L-spaces. A left-ordering on a group G is a total ordering ≺ on the elements of G that is invariant under left-multiplication; that is, g ≺ h implies fg ≺ fh for all f; g; h 2 G. A group is said to be left-orderable if it is nontrivial and admits a left ordering. A 3-manifold M is called orderable if π1(M) is arXiv:1911.11745v1 [math.GT] 26 Nov 2019 left-orderable. If M is a rational homology 3-sphere, then the rank of its Heegaard Floer homology is bounded below by the order of its first (integral) homology group. M is called an L-space if equality holds; that is, if rk HFd(M) = jH1(M; Z)j. This work is motivated by the following proposed connection between L-spaces and orderability, first conjectured by Boyer, Gordon, and Watson. 1 Conjecture 1.1 ([BGW13]). An irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. In [BGW13], this equivalence was shown to hold for all closed, connected, orientable, geometric three-manifolds that are non-hyperbolic. Knot groups are of particular interest due to the fact that knot complements are often hyperbolic and also because the L-space surgery interval for an L-space knot is known to be [2g − 1; 1); where g is the Seifert genus of the knot [OSz11]. We shall primarily focus on the (−2; 3; 7)-pretzel knot, which is an L- space knot. This knot has genus 5, and so if Conjecture 1.1 holds, one would expect the orderable surgeries to be precisely from slopes in the interval (−∞; 9): It has been shown in [Nie19] that surgery along any slope outside this interval always yields a non-orderable manifold. Moreover, Nie also showed that surgery along a slopes in (−, ) yields an orderable manifold for some sufficiently small > 0: In this work, we improve the result to: Theorem 1.2. Let X denote the exterior of the (−2; 3; 7)-pretzel knot. Then X(r) is orderable for all r 2 (−∞; 6): This leaves the slope interval [6; 9) as still unverified vis-`a-visConjecture 1.1. The proof of the main theorem follows the strategy developed by Culler and Dunfield in [CD18]. In particular, we compute a certain one-parameter family of PSL^2(R)-representations of the knot group, and from that, we conclude the existence of a curve on the translation extension locus associated to such representations. In fact, Culler and Dunfield used numerical methods to produce an image of the translation extension locus associated to the knot P (−2; 3; 7) (see Figure 3 in [CD18]). Theorem 1.2 is precisely the result one expects from that image. The (−2; 3; 7)-pretzel knot can be viewed as one of a family of twisted torus knots. In particular, consider the (3; 3k + 2) torus knot with m full twists about a pair of strands (pictured in Figure 1), which we denote by m 1 T3;3k+2. Notice that T3;5 is the pretzel knot P (−2; 3; 7): In section 5, we 1 consider possible extensions of our result to the family T3;3+2. Acknowledgements The author would like to thank Professors Zolt´anSzab´oand Peter Ozsv´ath for suggesting and encouraging work on this problem. This work was sup- ported by the NSF RTG grant DMS-1502424. 2 k m m Figure 1: The twisted torus knot T3;3k+2: Here the boxes represent k and m full twists of the strands passing through. 2 Background ~ In what follows, we shall denote by G the group PSL2(R) and by G we shall denote its universal cover PSL^2(R). 2.1 Reducible Representations Let K ⊆ S3 be a knot and denote by X the knot exterior. The abelianization ∼ map π1(X) ! H1(X; Z) = Z sends the meridian µ to a generator of the first homology. For any ζ 2 S1 ⊆ C (the complex unit circle), one can define a ∼ reducible representation ρζ : π1(X) ! PSU(1; 1) = G given by composing the abelianization map with: H1(X; Z) ! PSU(1; 1) ζ1=2 0 [µ] 7! ± 0 ζ−1=2 where ζ1=2 is any square root of ζ. 3 We are interested in finding paths of irreducible representations that are deformations of some such reducible representation. The following theorem gives necessary and sufficient conditions for this to happen: Theorem 2.1 (Lemma 4.8 and Proposition 10.3 of [HP05]). With K and ζ as above, ρζ may be deformed into a family of irreducible PSL2(C) represen- tations if and only if ζ is a root of the Alexander polynomial. In that case, the irreducible curve of characters is contained in PSU(2) on one side and ∼ in PSU(1; 1) = PSL2(R) on the other side. 2.2 The Translation Extension Locus Following [CD18], we give a description of the translation extension locus and ~ explain how it may be used to construct left-orderings. Let ρ : π1(X) ! G be a representation of the fundamental group of the knot exterior. As G acts on the circle via orientation-preserving transformations, G~ acts on R via orientation-preserving transformations as well and, in fact, can be viewed as a subgroup of Homeo+(R). More precisely, we are viewing R as the universal cover of S1 via the covering map x 7! e2πix so that, for instance, lifts of the identity matrix act on R by integral translations. We are interested in when such representations factor through the fundamental group of the filled manifold X(r), for then, by [BRW05], it will follow that X(r) will be left-orderable. Let us define the translation map trans : G~ ! R by gn(x) − x trans(g) := lim n!1 n for any x 2 R (note this definition does not depend on x). We call an element of G~ elliptic, parabolic, or hyperbolic if its image under the natural projection p : G~ ! G is elliptic, parabolic, or hyperbolic, respectively. Note ~ 1 that for any elliptic g 2 G, trans(g) is equal to 2π times the rotation angle corresponding to the action of p(g), up to additon of an integer. Now let us define the translation extension locus. For each representation ~ ρ : π1(X) ! G; we may consider its restriction to the peripheral subgroup: ~ ∼ 2 ρjπ1(@X) : π1(@X) ! G. As @X is a torus, π1(@X) = Z is abelian, and hence all (nontrivial) elements in the image of ρjπ1(@X) are of the same type (elliptic, parabolic, or hyperbolic). So we shall call ρ elliptic, parabolic, or hyperbolic if the image of the restriction to the peripheral subgroup consists 4 of elements of the respective type. Given ρ as above, we may consider the composition trans ◦ρjπ1(@X) : π1(@X) ! R. Because the peripheral subgroup is abelian, this is actually a homomorphism, and hence may be considered ∼ 1 ∼ 2 as an element of Hom(π1(@X); R) = H (@X; R) = R . If (µ, λ) are the natural meridian-longitude basis for H1(@X; R); then one may consider the dual basis (µ∗; λ∗) for H1(@X; R). Using this basis, we may consider the subset E of R2 corresponding to all elliptic and parabolic representations ~ ρ : π1(X) ! G: The translation extension locus is the closure of this set. We ¯ 2 denote it ELG~(X) := E ⊆ R . p Let r = q 2 Q with p; q relatively prime. Then γ = pµ + qλ is a primitive element of H1(@X; Z) and is represented by a simple closed curve 1 in @X. Let Lr denote the set of elements of H (@X; R) which vanish on γ. In the meridian-longitude-dual basis, this corresponds to the (a; b) 2 R2 such that ap + bq = 0. Graphically, this is the line through the origin in 2 p R with slope − q = −r. Suppose an element in ELG~(X) \ Lr comes from an elliptic representation ρ. Then, trans(ρ(µpλq)) = 0. As remarked above, the translation of an elliptic element corresponds to the rotation angle about its fixed point, and so, ρ(µpλq) = 1 2 G:~ It follows that ρ factors through p q ∼ π1(X)=hhµ λ ii = π1(X(r)): It follows by [BRW05] that X(r) is orderable if it is irreducible. In fact, Culler and Dunfield proved the following slightly more general result, for M any rational homology solid torus: Theorem 2.2 (Lemma 4.4 of [CD18]). Suppose M is a compact orientable irreducible 3-manifold with @M a torus, and assume the Dehn filling M(r) is irreducible. If Lr meets ELG~(M) at a nonzero point which is not parabolic or ideal, then M(r) is orderable. Remark. Here an ideal point is an element of E¯ n E so that the hypotheses of the theorem may be equivalently expressed as Lr intersecting ELG~(M) at a nonzero elliptic point.
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